## Abstract

Rough-strip energy dissipators (R-SEDs) can be arranged at the bend bottom of curved spillways to dissipate energy and divert flow for bend flow. Using the entropy weight and TOPSIS methods, a multi-criteria evaluation system was established for comprehensive energy dissipation and flow diversion effects of R-SEDs. Orthogonal tests and numerical simulation were conducted to analyze factors affecting these effects (average R-SED height, R-SED angle, R-SED spacing, bend width, bend centerline radius and discharge flow rate). It was found that bend width and bend centerline radius significantly affected R-SEDs' energy dissipation effects. Average R-SED height, R-SED spacing and bend centerline radius significantly affected R-SEDs' flow diversion effects. Bend width, average R-SED height and bend centerline radius significantly affected R-SEDs' combined effects of energy dissipation and flow diversion. Their energy dissipation effects were larger than the flow diversion effects. R-SEDs can effectively alleviate adverse hydraulic phenomena in curved spillways. With the recommended parameters, R-SEDs showed the best performance, with the energy dissipation rate increasing by 18.67% and the water surface superelevation coefficient decreasing by 26.14%. The accuracy of the multi-criteria evaluation system was verified. This study can provide a reference for the R-SED design of similar curved spillways.

## HIGHLIGHTS

R-SEDs can effectively alleviate adverse hydraulic phenomena in curved spillways.

A multi-criteria evaluation system was established to assess R-SEDs’ performance.

Factors affecting energy dissipation and flow diversion of R-SEDs were analyzed.

Energy dissipation rate and water surface superelevation coefficient were constructed.

Optimal parameters were recommended for the best performance of R-SEDs.

## LIST OF SYMBOLS

*h*Average R-SED height

*α*R-SED angle

*s*R-SED spacing

*w*Bend width

*R*Bend centerline radius

*Q*Discharge flow rate

*i*Slope along the course

*L*_{1}Length of the straight inlet section

*L*_{2}Length of the straight outlet section

*h*_{1}R-SED height at the concave bank

*h*_{2}R-SED height at the convex bank

*O*The center of the curvature of the bend

*δ*R-SED thickness

*H*Water depth at the measurement point

*H*_{0}Head over the weir

*C*_{0}Discharge coefficient of the weir

*P*_{1}Height of the weir

*w*_{1}Width of the flow diversion canal at the upstream weir

*η*Energy dissipation rate

*η*_{(i)}Energy dissipation rate under the

*i*th condition*TE*_{1}Total mechanical energy per unit weight of water in the upstream flow cross-section

*TE*_{2}Total mechanical energy per unit weight of water in the downstream flow cross-section

*SS*Water surface superelevation coefficient

*SS*_{(i)}Water surface superelevation coefficient under the

*i*th condition*σ*_{i}Standard deviation of the superelevation of transverse water surface (

*Δφ*) of all calculated sections under the*i*th condition*μ*_{i}Mean value of superelevation of transverse water surface of all calculated sections under the

*i*th condition*Δφ*_{ij}Difference between the water surface at the concave bank of the bend and the horizontal plane across the center at the

*j*th calculated section under the*i*th condition*w*_{i}Water surface width of the open channel based on the centerline water surface elevation under the

*i*th condition*Δφ*Transverse water surface superelevation of the bend

*W*_{j}Entropy weight

*p*_{ij}Proportion of the

*i*th evaluation object under the*j*th index*E*_{j}Entropy value of the

*j*th index- ,
Euclidean distance and between each object and the ideal solution

*C*Relative closeness

*P*Significance factor

- DF
Degree of freedom

- Adj SS
Adjusted sum of squared deviation from mean

- Adj MS
Adjusted mean squared error

*F*Chi-square test factor

*η**Membership degree of energy dissipation (

*η*)*SS**Membership degree of water surface superelevation coefficient (

*SS*)*k*Turbulent kinetic energy

*ε*Turbulent kinetic energy dissipation rate

*α*_{w}Volume fraction of water in the calculation area with respect to the calculation element

*α*_{α}Volume fraction of air in the calculation area with respect to the calculation element

*T*Turbulent kinetic energy

## INTRODUCTION

The spillway is a key component of reservoir discharge structures (Kells & Smith 1991; Zhang *et al.* 2015; Damarnegara *et al.* 2020), mainly employed for energy dissipation and flood control. Influenced by topographic and geological conditions, construction conditions and engineering economy, some spillways must have corners, thus forming bends at the corner (Seo & Shin 2018; Yang *et al.* 2019). The water flowing through the bend is called bend flow, which is different from straight-section flow. When water flows through a bend, centrifugal inertia forces are generated by the curved movement of the bend flow. Thus, the surface and bottom flow migrate to the concave and convex banks, respectively, forming a closed transverse circulation within the bend section (Seyedashraf & Akhtari 2015). This circulation is combined with the longitudinal flow, forming a spiral flow. This spiral flow moves forward in the mainstream direction (Johannesson & Parker 1989). In addition, the water depth increases at the concave bank of the bend while decreasing at the convex bank. This causes a transverse gradient of the water surface (Dietrich *et al.* 1979).

Since Thomson (1876) revealed the existence of both longitudinal and lateral motion in bend flow, studies on bend flow have focused on two aspects, i.e., hydraulic characteristics and engineering measures. The studies on the hydraulic characteristics of bend flow mainly focus on the water depth distribution (Bathurst & Hey 1979; Molls & Chaudhry 1995; Qin *et al.* 2016; Zhou *et al.* 2017; Maatooq & Hameed 2020), flow velocity distribution (De Vriend & Geldof 1983; Anwar 1986; Odgaard & Bergs 1988; Ye & McCorquodale 1998; Han *et al.* 2011; Vaghefi *et al.* 2015; Moncho-Esteve *et al.* 2018; Pradhan *et al.* 2018; Schreiner *et al.* 2018; Hu *et al.* 2019; Kim *et al.* 2020; Yan *et al.* 2020) and secondary flow evolution (Jin & Steffler 1993; Booij 2003; Huai *et al.* 2012; Ramamurthy *et al.* 2013; Engel & Rhoads 2016; Gu *et al.* 2016; Shaheed *et al.* 2021). These studies generally describe the characteristics and evolution of bend flow and provide a theoretical basis for further studies. Based on the hydraulic characteristics of bend flow, some engineering measures in the bend have been proposed using physical model tests or numerical simulation in order to improve the bend flow structure. For example, Zhang *et al.* (2015) arranged continuous guide walls at the central axis of the bend to improve the bend flow pattern. However, the height of the guide wall needs to be determined based on the water depth at the bend inlet. The guide wall only applies to the bend with a small centerline radius. Thus, the practicality needs to be improved. Yang *et al.* (2019) arranged permeable spur dikes at the concave bank of the bend to improve the flow pattern of ‘backwater at the concave bank’. The locations with permeable spur dikes are significantly prone to congestion, which is not conducive to the structural stability of permeable spur dikes during long-term operation. Ranjan *et al.* (2006) arranged vanes at the central axis of the bend to reduce the secondary flow intensity of the bend flow. However, the vanes do not have energy dissipation effects and cannot dissipate the excess energy of the bend flow. Martin-Vide *et al.* (2010) arranged ripraps at the concave bank of the bend to reduce the scouring of bend flow at the concave bank. However, this did not change the flow pattern of ‘increasing water depth at the concave bank and decreasing water depth at the convex bank’ in the bend. In this paper, the rough-strip energy dissipators (R-SEDs) were arranged at the bend bottom of the curved spillway. These R-SEDs are simple in shape and convenient in construction and have achieved better performance in improving the flow structure and energy dissipation in the bend (Zhang *et al.* 2022a, 2022b).

The study on R-SEDs was mainly based on the hydraulic model test of the curved spillway of the Yin'ejike 635 Reservoir in Xinjiang, China (the geometric scale was 1:50). The physical model test was completed in the curved spillway flume of the Xinjiang Key Laboratory of Water Conservancy Engineering Safety and Water Disaster Prevention, China. At a discharge flow rate of 800 m^{3}/s, the water depth and flow velocity at the concave and convex banks were significantly different and the bend flow was turbulent. To tackle these adverse flow patterns, R-SEDs have been arranged at the bend bottom and have shown high effectiveness in energy dissipation and flow diversion (Zhang *et al.* 2022b). Since then, the R-SEDs have been increasingly investigated. For example, Zhang *et al.* (2022a) mainly analyzed the results of the R-SEDs in the hydraulic model test of the curved spillway of the Yin'ejike 635 Reservoir using the single-factor test method. They only considered the effects of R-SED arrangement parameters on the energy dissipation and flow diversion effects of R-SEDs. The effects of engineering parameters of the curved spillway on these effects were not included. Zhang *et al.* (2022b) derived an equation for calculating the energy dissipation rate of R-SEDs through physical model tests, without considering the flow diversion effect of R-SEDs on bend flow. Thus, these studies still have some limitations.

Therefore, in this paper, the R-SEDs were arranged at the bend bottom of the curved spillway. Orthogonal tests and numerical simulation were performed to evaluate the R-SED arrangement parameters (average R-SED height *h*, R-SED angle *α*, R-SED spacing *s*) and the spillway engineering parameters (bend width *w*, bend centerline radius *R*, discharge flow rate *Q*) that affect the energy dissipation and diversion effects of R-SEDs. The energy dissipation rate, water surface superelevation coefficient and relative closeness were selected as response variables. Based on the entropy weight and technique for order performance by similarity to ideal solution (TOPSIS) methods, a multi-criteria evaluation system was established to analyze these parameters and determine their optimal values. Thus, an optimal parameter combination was recommended, i.e., *h* = 1.8 cm, *s* = 24 cm, *α* = 18°, *R* = 200 cm, *w* = 80 cm and *Q* = 20.0 L/s. This can provide a reference for the R-SED design of similar curved spillways. In addition, the established water surface superelevation coefficient calculation model is simple and easy to understand, facilitating its application in engineering design.

## METHODOLOGY

### Orthogonal test design

#### Test apparatus

*i*= 0.025) and the straight inlet section (

*L*

_{1}= 60 cm) can ensure the smooth flow of inlet water into the bend section. According to Table 2, nine bends of different sizes (i.e., nine parameter combinations (

*R*×

*w*), including 140 cm × 50 cm, 170 cm × 60 cm, 200 cm × 80 cm, 140 cm × 80 cm, 170 cm × 50 cm, 200 cm × 60 cm, 200 cm × 50 cm, 140 cm × 60 and 170 cm × 80 cm) were designed and prepared for the test, with a bend angle of 60°. The straight outlet section (

*L*

_{2}= 140 cm) can smoothly connect with the flow out of the bend. The water circulation system consists of a water pump, a rectangular water storage tank, a water measuring weir, a water storage reservoir and water flow diversion pipelines.

#### Test program

Orthogonal experimental design, referred to as orthogonal design, is a method to scientifically arrange and analyze multi-factor tests using orthogonal tables. It is one of the most commonly used experimental design methods. In the orthogonal design, representative points that meet the requirements of balance and orthogonality can be selected from the whole test and tested so as to obtain the overall stability of the development trend (Wang *et al.* 2022; Yuan *et al.* 2022).

Considering the influence of R-SED arrangement parameters and spillway engineering parameters on the energy dissipation and flow diversion effect of R-SEDs, six influencing factors were selected, including average R-SED height (*h*), R-SED spacing (*s*), R-SED angle (*α*), bend centerline radius (*R*), bend width (*w*) and discharge flow rate (*Q*). Based on the previous findings of installing R-SEDs (Zhang *et al.* 2022a, 2022b), three levels were selected for each factor: (−1), 0 and (+1). The orthogonal table *L*_{18}(3^{7}) was selected for the test design. The factors and levels of the orthogonal test are shown in Table 1. In addition, the orthogonal test does not consider the effect of interlevel interactions. Thus, the evaluation index is only affected by the factors' independent effects (i.e., main effects) and random errors (i.e., residuals).

Level . | h (cm)
. | s (cm)
. | α (°)
. | R (cm)
. | w (cm)
. | Q (L/s)
. |
---|---|---|---|---|---|---|

−1 | 1.2 | 18 | 18 | 140 | 50 | 20.0 |

0 | 1.5 | 24 | 22 | 170 | 60 | 22.5 |

+1 | 1.8 | 30 | 26 | 200 | 80 | 25.0 |

Level . | h (cm)
. | s (cm)
. | α (°)
. | R (cm)
. | w (cm)
. | Q (L/s)
. |
---|---|---|---|---|---|---|

−1 | 1.2 | 18 | 18 | 140 | 50 | 20.0 |

0 | 1.5 | 24 | 22 | 170 | 60 | 22.5 |

+1 | 1.8 | 30 | 26 | 200 | 80 | 25.0 |

Run . | h (cm)
. | s (cm)
. | α (°)
. | R (cm)
. | w (cm)
. | Q (L/s)
. |
---|---|---|---|---|---|---|

1 | 1.2 | 18 | 18 | 140 | 50 | 20.0 |

2 | 1.2 | 24 | 22 | 170 | 60 | 22.5 |

3 | 1.2 | 30 | 26 | 200 | 80 | 25.0 |

4 | 1.5 | 18 | 18 | 170 | 60 | 25.0 |

5 | 1.5 | 24 | 22 | 200 | 80 | 20.0 |

6 | 1.5 | 30 | 26 | 140 | 50 | 22.5 |

7 | 1.8 | 18 | 22 | 140 | 80 | 22.5 |

8 | 1.8 | 24 | 26 | 170 | 50 | 25.0 |

9 | 1.8 | 30 | 18 | 200 | 60 | 20.0 |

10 | 1.2 | 18 | 26 | 200 | 60 | 22.5 |

11 | 1.2 | 24 | 18 | 140 | 80 | 25.0 |

12 | 1.2 | 30 | 22 | 170 | 50 | 20.0 |

13 | 1.5 | 18 | 22 | 200 | 50 | 25.0 |

14 | 1.5 | 24 | 26 | 140 | 60 | 20.0 |

15 | 1.5 | 30 | 18 | 170 | 80 | 22.5 |

16 | 1.8 | 18 | 26 | 170 | 80 | 20.0 |

17 | 1.8 | 24 | 18 | 200 | 50 | 22.5 |

18 | 1.8 | 30 | 22 | 140 | 60 | 25.0 |

Run . | h (cm)
. | s (cm)
. | α (°)
. | R (cm)
. | w (cm)
. | Q (L/s)
. |
---|---|---|---|---|---|---|

1 | 1.2 | 18 | 18 | 140 | 50 | 20.0 |

2 | 1.2 | 24 | 22 | 170 | 60 | 22.5 |

3 | 1.2 | 30 | 26 | 200 | 80 | 25.0 |

4 | 1.5 | 18 | 18 | 170 | 60 | 25.0 |

5 | 1.5 | 24 | 22 | 200 | 80 | 20.0 |

6 | 1.5 | 30 | 26 | 140 | 50 | 22.5 |

7 | 1.8 | 18 | 22 | 140 | 80 | 22.5 |

8 | 1.8 | 24 | 26 | 170 | 50 | 25.0 |

9 | 1.8 | 30 | 18 | 200 | 60 | 20.0 |

10 | 1.2 | 18 | 26 | 200 | 60 | 22.5 |

11 | 1.2 | 24 | 18 | 140 | 80 | 25.0 |

12 | 1.2 | 30 | 22 | 170 | 50 | 20.0 |

13 | 1.5 | 18 | 22 | 200 | 50 | 25.0 |

14 | 1.5 | 24 | 26 | 140 | 60 | 20.0 |

15 | 1.5 | 30 | 18 | 170 | 80 | 22.5 |

16 | 1.8 | 18 | 26 | 170 | 80 | 20.0 |

17 | 1.8 | 24 | 18 | 200 | 50 | 22.5 |

18 | 1.8 | 30 | 22 | 140 | 60 | 25.0 |

*h*

_{1}) was designed to be larger than the height at the convex bank (

*h*

_{2}), i.e.,

*h*

_{1}>

*h*

_{2}. Therefore, R-SEDs had a trapezoidal longitudinal section. Average R-SED height (

*h*) is the average value of the R-SED height at the concave bank (

*h*

_{1}) and the height at the concave bank (

*h*

_{2}), i.e.,

*h*

*=*(

*h*

_{1}+

*h*

_{2})/2. R-SED spacing (

*s*) is the straight line distance between the center of two adjacent R-SEDs. R-SED angle (

*α*) is the angle between the R-SED centerline and the direction perpendicular to the bend centerline. Bend centerline radius (

*R*) is the distance between the bend centerline and the center of the curvature of the bend (

*O*). Bend width (

*w*) is the horizontal distance between the two banks of the spillway. Discharge flow rate (

*Q*) is the discharge flow rate of the spillway during regular operation. The schematic diagram of each parameter is shown in Figure 2. The orthogonal tests were designed according to Table 1. The detailed test program is shown in Table 2.

#### Measurement arrangement

Test measurement instruments were selected and arranged according to the Chinese standard ‘Calibration Method of Common Instruments for Hydraulic and River Model Test’ (SL 233-2016) (Ministry of Water Resources of China 2016).

- (a)
Water depth measurement

The water level measurement probe was used to measure the water depth with an accuracy of 0.1 mm. A total of 51 water depth measurement cross-sections were arranged along the spillway model, i.e., #0 ∼ #50. Each cross-section was arranged with 11 measurement points (i.e., A ∼ K). To reduce the disturbance of the sidewall to the flow, two near-bank measurement points (A and E) were located 1 cm from the sidewalls of the concave and convex banks, respectively.

- (b)
Flow velocity measurement

The flow velocity was measured using a Pitot tube, with an accuracy of 0.1 mm. A total of 13 measurement cross-sections (i.e., #0, #4, #8, #12, #16, #20, #24, #28, #32, #36, #40, #44 and #48) were selected as flow velocity measurement cross-sections. Six measurement points (i.e., A, C, E, G, I and K) were selected for each cross-section. The vertical measurement position of each measurement point was located at 2*H*/3 from the bottom (*H* is the water depth at the measurement point).

- (c)
Flow rate measurement

*Q*is the discharge flow rate (L/s);

*H*

_{0}is the head over the weir (m);

*C*

_{0}is the discharge coefficient of the weir, which is related to the size of the opening and can be calculated bywhere

*P*

_{1}is the height of the weir (m);

*w*

_{1}is the width of the flow diversion canal at the upstream weir (m).

### Construction of evaluation indexes

#### Energy dissipation rate

The R-SEDs have a dual effect of energy dissipation and flow diversion on the bend flow. Evaluation indexes need to be constructed to assess the energy dissipation effect and flow diversion effect of the R-SEDs in the 18 orthogonal tests.

*η*) was introduced as an evaluation index. The change interval of the energy dissipation rate is (0, 1), and a larger energy dissipation rate indicates a larger energy dissipation effect. The energy dissipation rate is calculated throughwhere

*η*

_{(i)}is the energy dissipation rate under the

*i*th condition (%);

*TE*

_{1}is the total mechanical energy per unit weight of water in the upstream flow cross-section (m);

*Z*

_{1}is the minimum elevation of the upstream flow cross-section (m);

*H*

_{1}is the average water depth of the upstream flow cross-section (m);

*v*

_{1}is the average velocity of the upstream flow cross-section (m/s);

*TE*

_{2}is the total mechanical energy per unit weight of water in the downstream flow cross-section (m);

*Z*

_{2}is the minimum elevation of the downstream flow cross-section (m);

*H*

_{2}is the average water depth of the downstream flow cross-section (m);

*v*

_{2}is the average velocity of the downstream flow cross-section (m/s);

*α*

_{0}is the kinetic energy correction coefficient (.

*α*

_{0}depends on the flow velocity distribution at the flow cross-section. For the gradually varied flow,

*α*

_{0}= 1.0–1.05 and is commonly taken as 1.0 in engineering practice. In this paper, the bend flow in the spillway belonged to a nonuniform gradually-varied flow. Thus,

*α*

_{0}was taken as 1.0) (Zhang

*et al.*2022a);

*g*is the acceleration of gravity, taken as 9.81 m/s

^{2}.

Cross-sections #8 and #32 were selected as the upstream and downstream sections of the bend, respectively. The horizontal plane where the bottom elevation of Cross-section #32 was located was taken as the reference plane. The energy dissipation rate in the 18 test scenarios was calculated using Equations (3)–(5). The calculation results of the energy dissipation rate are shown in Table 3.

Run . | h (cm)
. | s (cm)
. | α (°)
. | R (°)
. | w (cm)
. | Q (cm)
. | η (–)
. | SS (–)
. |
---|---|---|---|---|---|---|---|---|

1 | 1.2 | 18 | 18 | 140 | 50 | 20.0 | 0.305 | 0.466 |

2 | 1.2 | 24 | 22 | 170 | 60 | 22.5 | 0.407 | 0.359 |

3 | 1.2 | 30 | 26 | 200 | 80 | 25.0 | 0.415 | 0.241 |

4 | 1.5 | 18 | 18 | 170 | 60 | 25.0 | 0.358 | 0.488 |

5 | 1.5 | 24 | 22 | 200 | 80 | 20.0 | 0.498 | 0.386 |

6 | 1.5 | 30 | 26 | 140 | 50 | 22.5 | 0.278 | 0.414 |

7 | 1.8 | 18 | 22 | 140 | 80 | 22.5 | 0.354 | 0.555 |

8 | 1.8 | 24 | 26 | 170 | 50 | 25.0 | 0.322 | 0.456 |

9 | 1.8 | 30 | 18 | 200 | 60 | 20.0 | 0.429 | 0.489 |

10 | 1.2 | 18 | 26 | 200 | 60 | 22.5 | 0.396 | 0.328 |

11 | 1.2 | 24 | 18 | 140 | 80 | 25.0 | 0.434 | 0.408 |

12 | 1.2 | 30 | 22 | 170 | 50 | 20.0 | 0.346 | 0.272 |

13 | 1.5 | 18 | 22 | 200 | 50 | 25.0 | 0.332 | 0.384 |

14 | 1.5 | 24 | 26 | 140 | 60 | 20.0 | 0.314 | 0.467 |

15 | 1.5 | 30 | 18 | 170 | 80 | 22.5 | 0.445 | 0.367 |

16 | 1.8 | 18 | 26 | 170 | 80 | 20.0 | 0.413 | 0.523 |

17 | 1.8 | 24 | 18 | 200 | 50 | 22.5 | 0.360 | 0.438 |

18 | 1.8 | 30 | 22 | 140 | 60 | 25.0 | 0.276 | 0.497 |

Run . | h (cm)
. | s (cm)
. | α (°)
. | R (°)
. | w (cm)
. | Q (cm)
. | η (–)
. | SS (–)
. |
---|---|---|---|---|---|---|---|---|

1 | 1.2 | 18 | 18 | 140 | 50 | 20.0 | 0.305 | 0.466 |

2 | 1.2 | 24 | 22 | 170 | 60 | 22.5 | 0.407 | 0.359 |

3 | 1.2 | 30 | 26 | 200 | 80 | 25.0 | 0.415 | 0.241 |

4 | 1.5 | 18 | 18 | 170 | 60 | 25.0 | 0.358 | 0.488 |

5 | 1.5 | 24 | 22 | 200 | 80 | 20.0 | 0.498 | 0.386 |

6 | 1.5 | 30 | 26 | 140 | 50 | 22.5 | 0.278 | 0.414 |

7 | 1.8 | 18 | 22 | 140 | 80 | 22.5 | 0.354 | 0.555 |

8 | 1.8 | 24 | 26 | 170 | 50 | 25.0 | 0.322 | 0.456 |

9 | 1.8 | 30 | 18 | 200 | 60 | 20.0 | 0.429 | 0.489 |

10 | 1.2 | 18 | 26 | 200 | 60 | 22.5 | 0.396 | 0.328 |

11 | 1.2 | 24 | 18 | 140 | 80 | 25.0 | 0.434 | 0.408 |

12 | 1.2 | 30 | 22 | 170 | 50 | 20.0 | 0.346 | 0.272 |

13 | 1.5 | 18 | 22 | 200 | 50 | 25.0 | 0.332 | 0.384 |

14 | 1.5 | 24 | 26 | 140 | 60 | 20.0 | 0.314 | 0.467 |

15 | 1.5 | 30 | 18 | 170 | 80 | 22.5 | 0.445 | 0.367 |

16 | 1.8 | 18 | 26 | 170 | 80 | 20.0 | 0.413 | 0.523 |

17 | 1.8 | 24 | 18 | 200 | 50 | 22.5 | 0.360 | 0.438 |

18 | 1.8 | 30 | 22 | 140 | 60 | 25.0 | 0.276 | 0.497 |

#### Water surface superelevation coefficient

*SS*) was introduced as an evaluation index.

*SS*varied between (0, 1) and is negatively correlated with flow diversion effects, i.e., a smaller

*SS*indicates better flow diversion effects of the R-SED.

*SS*is expressed aswhere

*i*indicates the sequence number of the condition;

*j*indicates the sequence number of the calculated cross-section;

*SS*

_{(i)}is the water surface superelevation coefficient under the

*i*th condition;

*σ*is the standard deviation of the superelevation of transverse water surface (

_{i}*Δφ*) of all calculated sections under the

*i*th condition (m);

*μ*is the mean value of superelevation of transverse water surface of all calculated sections under the

_{i}*i*th condition (m);

*Δφ*is the difference between the water surface at the concave bank of the bend and the horizontal plane across the center at the

_{ij}*j*th calculated cross-section under the

*i*th condition (m);

*k*

_{0}is the superelevation coefficient, taken as 0.5 for a simple circular-curved bend of a rectangular open channel;

*v*is the average velocity of the

_{ij}*j*th calculated section under the

*i*th condition (m/s);

*w*is the water surface width of the open channel based on the centerline water surface elevation under the

_{i}*i*th condition (m);

*g*is the acceleration of gravity, taken as 9.81 m/s

^{2};

*r*is the bend centerline radius under the

_{i}*i*th condition (m). The calculation of the transverse water surface superelevation of the bend (

*Δφ*) is illustrated in Figure 5.

To fully measure the flow diversion effect of R-SEDs, Cross-sections #8 ∼ #48 (41 cross-sections in total) were selected to calculate the water surface superelevation coefficient. The water surface superelevation coefficients were calculated using Equations (6)–(9) for 18 orthogonal tests. The calculation results are shown in Table 3.

### Multi-criteria evaluation based on entropy weight method and TOPSIS method

- (a)
Establishing an evaluation system of the combined effect of energy dissipation and flow diversion

Eighteen groups of orthogonal tests were used as feasibility study schemes, and the energy dissipation rate and the water surface superelevation coefficient were used as target variables to construct the original matrix .

- (b)
Determining the weight of the energy dissipation rate and the water surface superelevation coefficient using the entropy weight method

*et al.*2021). The matrix

*Y*was derived from the nondimensionalization of the original matrix and then the entropy weight

*W*of the energy dissipation rate and the water surface superelevation coefficient was determined. A larger entropy weight indicates that the evaluation index is more important. The calculation formula is expressed aswhere

_{j}*r*indicates the

_{ij}*j*th evaluation index of the

*i*th evaluation objects;

*i*denotes the number of evaluation objects (

*i*= 1, 2, … ,

*m*);

*j*denotes the number of evaluation indexes (

*j*= 1, 2, … ,

*n*);

*m*evaluation objects refer to

*m*test runs, and each test run is an evaluation object;

*n*evaluation indexes refer to the energy dissipation rate and the water surface superelevation coefficient;

*p*is the proportion of the

_{ij}*i*th evaluation object under the

*j*th index;

*E*is the entropy value of the

_{j}*j*th index.

- (c)
Obtaining the overall ranking of each program using the TOPSIS method

*et al.*2019). Firstly, the original matrix is normalized to obtain a matrix and the matrix

*B*is multiplied by the entropy weight

*W*to obtain the weighting matrix

_{j}*Z*. Secondly, the Euclidean distance and between each object and the ideal solution as well as the relative closeness

*C*of each object is calculated. A larger relative closeness indicates that the solution is closer to the ideal solution and that the rating is better. Finally, the solutions are comprehensively ranked according to the relative closeness of each solution to form a decision basis. The calculation equations are as follows:where

*z* j*is the optimal solution (positive ideal solution) for the

*j*th evaluation index;

*z- j*is the worst solution (negative ideal solution) for the

*j*th evaluation index.

In summary, the established multi-criteria evaluation system consists of two sub-evaluation indexes (the energy dissipation rate and water surface superelevation coefficient) and one comprehensive evaluation index (relative closeness).

### Model development

#### Modeling and meshing

Firstly, a three-dimensional (3D) spillway model was established using SOLIDWORKS 2018. The 3D model included the straight inlet section, the 60° bend and the straight outlet section, as shown in Figure 1(b). Then, the calculation area of the spillway was extracted and meshed using ANSYS ICEM CFD 19.0. Considering the complex bend structure and free water surface in the spillway, the hybrid structured meshes (tetrahedral and hexahedral elements) were used to mesh the 3D calculation area. The structured meshes have the advantages of arbitrary structure and strong adaptability. Finally, the spillway model was calculated using ANSYS FLUENT 19.0.

^{5}, 8 × 10

^{5}, 9 × 10

^{5}, 1.0 × 10

^{6}and 1.1 × 10

^{6}) were set for the geometric spillway model in Run 5, respectively. Then, the spillways with the five different mesh numbers were calculated using the same meshing and simulation methods, respectively. The simulated water depth and flow velocity of the spillway with five different mesh quantities were extracted and compared with the measured values from the physical model test. The comparison results are shown in Figure 6. The differences between simulated and experimental values were the most significant at 7 × 10

^{5}mesh elements. Then, the difference decreased with the number of mesh elements increasing to 8 × 10

^{5}. The test results were consistent with the simulation results at 9 × 10

^{5}, 1.0 × 10

^{6}and 1.1 × 10

^{6}mesh elements. The simulation results for the water surface structure of the spillway at the three mesh element numbers (9 × 10

^{5}, 1.0 × 10

^{6}and 1.1 × 10

^{6}) were generally the same (Figure 6(c)). Therefore, the number of mesh elements was selected as 9 × 10

^{5}for the spillway model. The mesh distribution is shown in Figure 7.

#### Turbulence model and control equations

The renormalization group (RNG) *k*-*ɛ* turbulence model was selected as the turbulence model. Derived from a rigorous statistical technique, this model considers turbulent vortices and can accelerate the calculation of additional terms of the turbulent kinetic energy dissipation rate equation. It has successfully simulated many complex flow problems (Huai *et al.* 2012; Ramamurthy *et al.* 2013; Ghazanfari-Hashemi *et al.* 2019). The two equations of the RNG *k*-*ε* turbulence model are as follows:

*ε*) equation:andwhere

*k*is the turbulent kinetic energy;

*ε*is the turbulent kinetic energy dissipation rate ;

*μ*is the hydrodynamic viscosity;

*μ*

_{eff}is the effective viscosity coefficient;

*μ*

_{t}is the turbulent viscosity;

*ρ*is the volume fraction averaged density;

*α*and

_{k}*α*are the inverse Prandtl number for

_{ε}*k*and

*ε*, respectively;

*G*represents the generation of turbulent kinetic energy

_{k}*k*due to the mean flow velocity gradient;

*E*is the time-averaged strain rate of the fluid;

_{ij}*C*

_{1ε},

*C*

_{2ε}and

*C*are empirical coefficients;

_{μ}*t*is time; in each of the above equations,

*i*= 1, 2, 3, i.e., and ;

*j*is the summation subscript. The values of the model constants in the above equations are shown in Table 4.

α
. _{k} | α
. _{ε} | C_{1ε}
. | C_{2ε}
. | C
. _{μ} | η_{0}
. | β^{′}
. |
---|---|---|---|---|---|---|

1.39 | 1.39 | 1.42 | 1.68 | 0.0845 | 4.377 | 0.012 |

α
. _{k} | α
. _{ε} | C_{1ε}
. | C_{2ε}
. | C
. _{μ} | η_{0}
. | β^{′}
. |
---|---|---|---|---|---|---|

1.39 | 1.39 | 1.42 | 1.68 | 0.0845 | 4.377 | 0.012 |

#### Multi-phase flow model and boundary condition setting

*et al.*2020; Li

*et al.*2022). The principle of the VOF method is to define the functions

*α*(

_{w}*x*,

*y*,

*z*,

*t*) and

*α*(

_{a}*x*,

*y*,

*z*,

*t*) as the volume fraction of water and air in the calculation area with respect to the calculation element, respectively. The sum of the water and air volume fraction in each calculation element was equal to 1 (i.e.,

*α*+

_{w}*α*= 1). The control equation of

_{a}*α*is shown in Equation (22). Then, the location of the water–air interface can be detected by solving the following equation.where

_{w}*u*is the velocity component,

_{i}*x*is the coordinate component and

_{i}*t*is the time.

In this study, numerical simulations of the spillway flow were performed using the RNG *k*-*ε* turbulent flow model and the VOF method. The control volume finite element method (CVFEM) method was used to discretize the control equations. The semi-implicit SIMPLE algorithm was used to solve the velocity-pressure field coupling. PRESTO! was selected to calculate the pressure equations. The standard wall function method was used to deal with the near-wall flow.

The boundary condition setting of the spillway model is shown in Figure 7. According to the model test results, the water-air interface was set at the inlet water depth of 0.056 m. Thus, the water and air inlets were delineated. The boundary conditions of the water and air inlets were set as the velocity inlet and the pressure inlet (1 atm). The outlet boundary condition was set as the pressure outlet (1 atm). The standard no-slip solid wall boundary condition was applied to the remaining solid boundaries. The initial inlet water depth and flow velocity were 0.056 m and 0.448 m/s, respectively.

### Statistical analysis of data

Minitab 21.1 software was used to perform the analysis of variance (ANOVA) and intuitive analysis on the calculated energy dissipation rate, water surface superelevation coefficient and relative closeness for the 18 orthogonal tests.

## RESULTS AND DISCUSSION

### Energy dissipation rate

#### Analysis of variance

Using the energy dissipation rate as the response variable, ANOVA was conducted on the six influencing factors (predictor variables). The results are shown in Table 5. The significance level was selected as 0.05. When the *P*-value corresponding to the main effect was less than 0.05 (*P* < 0.05), the original hypothesis was rejected, and the total effect of the regression was considered significant. The analysis was focused on the main effects on the evaluation indexes without considering the interactions between different factors. Adjusted *R*^{2} (*R*^{2} (adj)) = 83.98% > 80.0% (Table 5) for the influencing factors, indicating that the orthogonal tests were statistically significant. Further analysis of the test levels shows that the differences between the different groups of bend centerline radius (*R*) and bend width (*w*) were significant (*P _{R}* < 0.05,

*P*< 0.05). This indicates that these two factors dominated the changing trend of the energy dissipation rate. The remaining factors had a test level of

_{w}*P*> 0.05, indicating that these factors showed homogeneity of variance and did not exhibit significant differences. This indicates that the remaining factors had insignificant effects on the energy dissipation rate. The above analysis shows that the spillway engineering parameters influenced the energy dissipation rate more than the R-SED arrangement parameters. Therefore, the influence of the spillway body shape on the energy dissipation rate of R-SEDs should be highlighted in engineering design.

Source . | DF . | Adj SS . | Adj MS . | F-value
. | P-value
. | Significance . |
---|---|---|---|---|---|---|

h | 2 | 0.001851 | 0.000926 | 1.51 | 0.306 | – |

s | 2 | 0.002978 | 0.001489 | 2.43 | 0.183 | – |

α | 2 | 0.003155 | 0.001578 | 2.58 | 0.170 | – |

R | 2 | 0.019343 | 0.009672 | 15.80 | 0.007 | * |

w | 2 | 0.032181 | 0.016091 | 26.28 | 0.002 | * |

Q | 2 | 0.002392 | 0.001196 | 1.95 | 0.236 | – |

Error | 5 | 0.003061 | 0.000612 | – | – | – |

Total | 17 | 0.064963 | – | – | – | – |

R^{2} = 0.9529 | R^{2} (adj) = 0.8398 |

Source . | DF . | Adj SS . | Adj MS . | F-value
. | P-value
. | Significance . |
---|---|---|---|---|---|---|

h | 2 | 0.001851 | 0.000926 | 1.51 | 0.306 | – |

s | 2 | 0.002978 | 0.001489 | 2.43 | 0.183 | – |

α | 2 | 0.003155 | 0.001578 | 2.58 | 0.170 | – |

R | 2 | 0.019343 | 0.009672 | 15.80 | 0.007 | * |

w | 2 | 0.032181 | 0.016091 | 26.28 | 0.002 | * |

Q | 2 | 0.002392 | 0.001196 | 1.95 | 0.236 | – |

Error | 5 | 0.003061 | 0.000612 | – | – | – |

Total | 17 | 0.064963 | – | – | – | – |

R^{2} = 0.9529 | R^{2} (adj) = 0.8398 |

*Note*: DF is the degree of freedom; Adj SS is the adjusted sum of squared deviation from mean; Adj MS is the adjusted mean squared error; *F* is the chi-square test factor; *P* is the significance factor.

**P* < 0.05, and the regression is significant

#### Intuitive analysis

*w*) > bend centerline radius (

*R*) > R-SED angle (

*α*) > R-SED spacing (

*s*) > discharge flow rate (

*Q*) > average R-SED height (

*h*). This shows the degree of influence of each factor on the energy dissipation rate. Thus, for the R-SED design, the effects of the significant factors (bend width

*w*and bend centerline radius

*R*) on the energy dissipation rate should be mainly considered. The values of these significant factors can be selected based on engineering characteristics and the main effect analysis of the energy dissipation rate (Figure 8). The remaining factors (secondary factors) can be selected based on the main effect analysis of the energy dissipation rate (Figure 8) or engineering experience.

Level . | h (cm)
. | s (cm)
. | α (°)
. | R (cm)
. | w (cm)
. | Q (L/s)
. |
---|---|---|---|---|---|---|

1 | 0.3838 | 0.3597 | 0.3885 | 0.3268 | 0.3238 | 0.3842 |

2 | 0.3708 | 0.3892 | 0.3688 | 0.3818 | 0.3633 | 0.3733 |

3 | 0.3590 | 0.3648 | 0.3563 | 0.4050 | 0.4265 | 0.3562 |

Delta | 0.0248 | 0.0295 | 0.0322 | 0.0782 | 0.1027 | 0.0280 |

Rank | 6 | 4 | 3 | 2 | 1 | 5 |

Level . | h (cm)
. | s (cm)
. | α (°)
. | R (cm)
. | w (cm)
. | Q (L/s)
. |
---|---|---|---|---|---|---|

1 | 0.3838 | 0.3597 | 0.3885 | 0.3268 | 0.3238 | 0.3842 |

2 | 0.3708 | 0.3892 | 0.3688 | 0.3818 | 0.3633 | 0.3733 |

3 | 0.3590 | 0.3648 | 0.3563 | 0.4050 | 0.4265 | 0.3562 |

Delta | 0.0248 | 0.0295 | 0.0322 | 0.0782 | 0.1027 | 0.0280 |

Rank | 6 | 4 | 3 | 2 | 1 | 5 |

The characteristic mean of each factor level is first obtained. Then, the main effect plot can be achieved by connecting the response mean of each factor level using a line. Whether a factor has the main effect can be determined by comparing the slope of the regression lines. A larger slope of the regression line indicates more significant main effects. The main effect plot of the energy dissipation rate (*η*) is shown in Figure 8. The regression lines of bend width (*w*) and bend centerline radius (*R*) were the steepest, indicating that *η* was the most sensitive to the change of these two factors. The two factors (*w* and *R*) were positive indicators, i.e., at a larger factor, *η* is higher, and R-SEDs' energy dissipation effects are larger. *η* was the highest (0.427) at the largest bend width (*w* = 80 cm) while decreasing by 31.38% with *w* decreasing to 50 cm. *η* was the highest (0.405) at the largest bend centerline radius (*R* = 200 cm) while decreasing by 23.85% with *R* decreasing to 140 cm. The slopes of the regression lines of three factors (average R-SED height *h*, R-SED angle *α* and discharge flow rate *Q*) were similar, i.e., the effect of these three factors on *η* was consistent. These three factors were negative indicators, i.e., at a larger factor, *η* is lower, and R-SEDs' energy dissipation effects are lower. *η* was the highest (0.385) at the smallest average R-SED height (*h* = 1.2 cm) while decreasing by 7.54% with *h* increasing to 1.8 cm. *η* was the highest (0.388) at the smallest R-SED angle (*α* = 18°) while decreasing by 8.99% with *α* increasing to 26°. *η* was the highest (0.385) at the smallest discharge flow rate (*Q* = 20 L/s) while decreasing by 8.15% with *Q* increasing to 25.0 L/s. R-SED spacing (*s*) was a two-way indicator, i.e., *s* is a positive indicator within 18–24 cm and a negative indicator within 24–30 cm. *η* was the highest (0.389) at *s* = 24 cm. *η* increased by 8.06% with *s* increasing from 18 to 24 cm and increased by 6.28% with *s* decreasing from 30 to 24 cm.

### Water surface superelevation coefficient

#### Analysis of variance

The water surface superelevation coefficient was used as the response variable. The significance level was selected as 0.05. The ANOVA was performed on the six influencing factors. The results are shown in Table 7. *R*^{2} (adj) = 92.08% > 80.0% for these influencing factors (Table 7), indicating that the orthogonal test was statistically significant. Further analysis shows that the test levels of average R-SED height (*h*), R-SED spacing (*s*) and bend centerline radius (*R*) differed significantly between groups (*P _{h}* < 0.05,

*P*< 0.05 and

_{s}*P*< 0.05). This indicates that these three factors dominated the changing trend of the water surface superelevation coefficient. The remaining factors had a test level of

_{R}*P*> 0.05. Thus, these factors had homogeneity of variance and did not show significant differences. Thus, they had fewer effects on the water surface superelevation coefficient. The above analysis shows that compared with the spillway engineering parameters, the R-SED arrangement parameters significantly affected the water surface superelevation coefficient.

Source . | DF . | Adj SS . | Adj MS . | F-value
. | P-value
. | Significance . |
---|---|---|---|---|---|---|

h | 2 | 0.065132 | 0.032566 | 57.48 | 0.000 | * |

s | 2 | 0.017942 | 0.008971 | 15.83 | 0.007 | * |

α | 2 | 0.005184 | 0.002592 | 4.58 | 0.074 | – |

R | 2 | 0.024958 | 0.012479 | 22.03 | 0.003 | * |

w | 2 | 0.003534 | 0.001767 | 3.12 | 0.132 | – |

Q | 2 | 0.002054 | 0.001027 | 1.81 | 0.256 | – |

Error | 5 | 0.002833 | 0.000567 | – | – | – |

Total | 17 | 0.121637 | – | – | – | – |

R^{2} = 0.9767 | R^{2} (adj) = 0.9208 |

Source . | DF . | Adj SS . | Adj MS . | F-value
. | P-value
. | Significance . |
---|---|---|---|---|---|---|

h | 2 | 0.065132 | 0.032566 | 57.48 | 0.000 | * |

s | 2 | 0.017942 | 0.008971 | 15.83 | 0.007 | * |

α | 2 | 0.005184 | 0.002592 | 4.58 | 0.074 | – |

R | 2 | 0.024958 | 0.012479 | 22.03 | 0.003 | * |

w | 2 | 0.003534 | 0.001767 | 3.12 | 0.132 | – |

Q | 2 | 0.002054 | 0.001027 | 1.81 | 0.256 | – |

Error | 5 | 0.002833 | 0.000567 | – | – | – |

Total | 17 | 0.121637 | – | – | – | – |

R^{2} = 0.9767 | R^{2} (adj) = 0.9208 |

**P* < 0.05, and the regression is significant

#### Intuitive analysis

*h*) > bend centerline radius (

*R*) > R-SED spacing (

*s*) > R-SED angle (

*α*) > bend width (

*w*) > discharge flow rate (

*Q*). This shows the degree of influence of each factor on the water surface superelevation coefficient. Thus, for the R-SED design, the effects of the significant factors (average R-SED height (

*h*), bend centerline radius (

*R*) and R-SED spacing (

*s*)) on water surface superelevation coefficient should be mainly considered. The values of these significant factors can be selected based on engineering characteristics and the main effect analysis of the water surface superelevation coefficient (Figure 9). The remaining factors (secondary factors) can be selected based on the main effect analysis of

*SS*(Figure 9) or engineering experience.

Level . | h (cm)
. | s (cm)
. | α (°)
. | R (cm)
. | w (cm)
. | Q (L/s)
. |
---|---|---|---|---|---|---|

1 | 0.3457 | 0.4573 | 0.4427 | 0.4678 | 0.4050 | 0.4338 |

2 | 0.4177 | 0.4190 | 0.4088 | 0.4108 | 0.4380 | 0.4102 |

3 | 0.4930 | 0.3800 | 0.4048 | 0.3777 | 0.4133 | 0.4123 |

Delta | 0.1473 | 0.0773 | 0.0378 | 0.0902 | 0.0330 | 0.0237 |

Rank | 1 | 3 | 4 | 2 | 5 | 6 |

Level . | h (cm)
. | s (cm)
. | α (°)
. | R (cm)
. | w (cm)
. | Q (L/s)
. |
---|---|---|---|---|---|---|

1 | 0.3457 | 0.4573 | 0.4427 | 0.4678 | 0.4050 | 0.4338 |

2 | 0.4177 | 0.4190 | 0.4088 | 0.4108 | 0.4380 | 0.4102 |

3 | 0.4930 | 0.3800 | 0.4048 | 0.3777 | 0.4133 | 0.4123 |

Delta | 0.1473 | 0.0773 | 0.0378 | 0.0902 | 0.0330 | 0.0237 |

Rank | 1 | 3 | 4 | 2 | 5 | 6 |

The main effect plot of water surface superelevation coefficient (*SS*) is shown in Figure 9. The regression lines of average R-SED height (*h*), R-SED spacing (*s*) and bend centerline radius (*R*) were the steepest. This indicates that *SS* was the most sensitive to the changes in these three factors. Average R-SED height (*h*) was a negative indicator, i.e., at a larger *h*, *SS* is higher, and R-SEDs' flow diversion effects are lower. *SS* was the smallest (0.345) at the smallest average R-SED height (*h* = 1.2 cm) while increasing by 29.88% with *h* increasing to 1.8 cm. R-SED spacing (*s*), bend centerline radius (*R*) and R-SED angle (*α*) were positive indicators, i.e., at a larger value, *SS* is smaller, and R-SEDs' flow diversion effects are larger. *SS* was the smallest (0.388) at the largest R-SED spacing (*s* = 30 cm) while increasing by 15.47% with *s* decreasing to 18 cm. *SS* was the smallest (0.378) at the largest bend centerline radius (*R* = 200 cm) while increasing by 19.40% with *R* decreasing to 140 cm. *SS* was the smallest (0.406) at the largest R-SED angle (*α* = 26°) while increasing by 8.56% with *α* decreasing to 18°. Bend width (*w*) and discharge flow rate (*Q*) were two-way indicators. The bend width (*w*) was a negative indicator within 50–60 cm and a positive indicator within 60–80 cm. *SS* is the largest at *w* = 60 cm, with the weakest flow diversion effect. *SS* decreased by 7.27% with *w* decreasing from 60 to 50 cm while decreasing by 5.68% from 60 to 80 cm, with improved flow diversion effects. The discharge flow rate (*Q*) was a positive indicator within 20.0–22.5 L/s and a negative indicator within 22.5–25.0 L/s. At *Q**=* 22.5 L/s, *SS* was the smallest (0.409), and the R-SEDs had the most significant flow diversion effect.

### Relative closeness

#### Determination of the weights of each evaluation index using the entropy weight method

The weights of each evaluation index (*η* and *SS*) were calculated using Equations (10)–(13) and the calculated parameters are shown in Table 9. The weight of the energy dissipation rate (*W*_{(η)} = 0.629) was larger than that of the superelevation coefficient (*W*_{(SS)} = 0.371). This indicates that the energy dissipation rate carried more information than the water surface superelevation coefficient and that the energy dissipation effect of R-SEDs on the bend flow was larger than the flow diversion effect.

Run . | η
. | SS
. | η*
. | SS*
. | p_{(η*)}
. | p_{(SS*)}
. | E_{(η)}
. | E_{(SS)}
. | W_{(η)}
. | W_{(SS)}
. |
---|---|---|---|---|---|---|---|---|---|---|

1 | 0.305 | 0.466 | 0.131 | 0.717 | 0.017 | 0.070 | 0.917 | 0.951 | 0.629 | 0.371 |

2 | 0.407 | 0.359 | 0.590 | 0.376 | 0.076 | 0.037 | ||||

3 | 0.415 | 0.241 | 0.626 | 0.000 | 0.081 | 0.000 | ||||

4 | 0.358 | 0.488 | 0.369 | 0.787 | 0.048 | 0.077 | ||||

5 | 0.498 | 0.386 | 1.000 | 0.462 | 0.130 | 0.045 | ||||

6 | 0.278 | 0.414 | 0.009 | 0.551 | 0.001 | 0.054 | ||||

7 | 0.354 | 0.555 | 0.351 | 1.000 | 0.046 | 0.098 | ||||

8 | 0.322 | 0.456 | 0.207 | 0.685 | 0.027 | 0.067 | ||||

9 | 0.429 | 0.489 | 0.689 | 0.790 | 0.089 | 0.078 | ||||

10 | 0.396 | 0.328 | 0.541 | 0.277 | 0.070 | 0.027 | ||||

11 | 0.434 | 0.408 | 0.712 | 0.532 | 0.092 | 0.052 | ||||

12 | 0.346 | 0.272 | 0.315 | 0.099 | 0.041 | 0.010 | ||||

13 | 0.332 | 0.384 | 0.252 | 0.455 | 0.033 | 0.045 | ||||

14 | 0.314 | 0.467 | 0.171 | 0.720 | 0.022 | 0.071 | ||||

15 | 0.445 | 0.367 | 0.761 | 0.401 | 0.099 | 0.039 | ||||

16 | 0.413 | 0.523 | 0.617 | 0.898 | 0.080 | 0.088 | ||||

17 | 0.36 | 0.438 | 0.378 | 0.627 | 0.049 | 0.062 | ||||

18 | 0.276 | 0.497 | 0.000 | 0.815 | 0.000 | 0.080 |

Run . | η
. | SS
. | η*
. | SS*
. | p_{(η*)}
. | p_{(SS*)}
. | E_{(η)}
. | E_{(SS)}
. | W_{(η)}
. | W_{(SS)}
. |
---|---|---|---|---|---|---|---|---|---|---|

1 | 0.305 | 0.466 | 0.131 | 0.717 | 0.017 | 0.070 | 0.917 | 0.951 | 0.629 | 0.371 |

2 | 0.407 | 0.359 | 0.590 | 0.376 | 0.076 | 0.037 | ||||

3 | 0.415 | 0.241 | 0.626 | 0.000 | 0.081 | 0.000 | ||||

4 | 0.358 | 0.488 | 0.369 | 0.787 | 0.048 | 0.077 | ||||

5 | 0.498 | 0.386 | 1.000 | 0.462 | 0.130 | 0.045 | ||||

6 | 0.278 | 0.414 | 0.009 | 0.551 | 0.001 | 0.054 | ||||

7 | 0.354 | 0.555 | 0.351 | 1.000 | 0.046 | 0.098 | ||||

8 | 0.322 | 0.456 | 0.207 | 0.685 | 0.027 | 0.067 | ||||

9 | 0.429 | 0.489 | 0.689 | 0.790 | 0.089 | 0.078 | ||||

10 | 0.396 | 0.328 | 0.541 | 0.277 | 0.070 | 0.027 | ||||

11 | 0.434 | 0.408 | 0.712 | 0.532 | 0.092 | 0.052 | ||||

12 | 0.346 | 0.272 | 0.315 | 0.099 | 0.041 | 0.010 | ||||

13 | 0.332 | 0.384 | 0.252 | 0.455 | 0.033 | 0.045 | ||||

14 | 0.314 | 0.467 | 0.171 | 0.720 | 0.022 | 0.071 | ||||

15 | 0.445 | 0.367 | 0.761 | 0.401 | 0.099 | 0.039 | ||||

16 | 0.413 | 0.523 | 0.617 | 0.898 | 0.080 | 0.088 | ||||

17 | 0.36 | 0.438 | 0.378 | 0.627 | 0.049 | 0.062 | ||||

18 | 0.276 | 0.497 | 0.000 | 0.815 | 0.000 | 0.080 |

*Note*: *η** is the degree of membership function of energy dissipation (*η*); *SS** is the degree of membership function of water surface superelevation coefficient (*SS*). The degree of membership is the ratio of (the index value − the minimum index value) to (the maximum index value − the minimum index value).

#### Determination of the overall ranking of each program using the TOPSIS method

The relative closeness can comprehensively measure the energy dissipation and flow diversion effects of R-SEDs. The relative closeness of each run to the ideal solution was calculated using Equations (14)–(18). Then, the overall ranking of the energy dissipation and flow diversion effect of R-SEDs on the bend flow at each run was obtained (Table 10). Run 5 (*h* = 1.5 cm, *s* = 24 cm, *α* = 22°, *R* = 200 cm, *w* = 80 cm and *Q* = 20L/s) had the largest relative closeness and the highest overall rating.

Run . | η
. | SS
. | 1 − SS
. | b_{(η)}
. | b_{(1−SS)}
. | z_{(η)}
. | z_{(1−SS)}
. | . | . | C
. | Rank . |
---|---|---|---|---|---|---|---|---|---|---|---|

1 | 0.305 | 0.466 | 0.534 | 0.191 | 0.214 | 0.120 | 0.080 | 0.083 | 0.032 | 0.279 | 16 |

2 | 0.407 | 0.359 | 0.641 | 0.255 | 0.198 | 0.160 | 0.074 | 0.054 | 0.057 | 0.514 | 7 |

3 | 0.415 | 0.241 | 0.759 | 0.260 | 0.133 | 0.164 | 0.049 | 0.072 | 0.055 | 0.432 | 11 |

4 | 0.358 | 0.488 | 0.512 | 0.224 | 0.270 | 0.141 | 0.100 | 0.057 | 0.060 | 0.513 | 8 |

5 | 0.498 | 0.386 | 0.614 | 0.312 | 0.213 | 0.196 | 0.079 | 0.035 | 0.092 | 0.728 | 1 |

6 | 0.278 | 0.414 | 0.586 | 0.174 | 0.229 | 0.110 | 0.085 | 0.091 | 0.035 | 0.279 | 16 |

7 | 0.354 | 0.555 | 0.445 | 0.222 | 0.307 | 0.140 | 0.114 | 0.057 | 0.071 | 0.557 | 6 |

8 | 0.322 | 0.456 | 0.544 | 0.202 | 0.252 | 0.127 | 0.093 | 0.072 | 0.048 | 0.397 | 12 |

9 | 0.429 | 0.489 | 0.511 | 0.269 | 0.270 | 0.169 | 0.100 | 0.030 | 0.079 | 0.722 | 2 |

10 | 0.396 | 0.328 | 0.672 | 0.248 | 0.181 | 0.156 | 0.067 | 0.061 | 0.051 | 0.451 | 10 |

11 | 0.434 | 0.408 | 0.592 | 0.272 | 0.225 | 0.171 | 0.084 | 0.039 | 0.071 | 0.644 | 4 |

12 | 0.346 | 0.272 | 0.728 | 0.217 | 0.150 | 0.136 | 0.056 | 0.083 | 0.028 | 0.254 | 18 |

13 | 0.332 | 0.384 | 0.616 | 0.208 | 0.212 | 0.131 | 0.079 | 0.074 | 0.037 | 0.331 | 15 |

14 | 0.314 | 0.467 | 0.533 | 0.197 | 0.258 | 0.124 | 0.096 | 0.075 | 0.049 | 0.394 | 13 |

15 | 0.445 | 0.367 | 0.633 | 0.279 | 0.203 | 0.175 | 0.075 | 0.044 | 0.071 | 0.620 | 5 |

16 | 0.413 | 0.523 | 0.477 | 0.259 | 0.289 | 0.163 | 0.107 | 0.034 | 0.079 | 0.698 | 3 |

17 | 0.36 | 0.438 | 0.562 | 0.226 | 0.242 | 0.142 | 0.090 | 0.059 | 0.052 | 0.468 | 9 |

18 | 0.276 | 0.497 | 0.503 | 0.173 | 0.274 | 0.109 | 0.102 | 0.088 | 0.052 | 0.372 | 14 |

Run . | η
. | SS
. | 1 − SS
. | b_{(η)}
. | b_{(1−SS)}
. | z_{(η)}
. | z_{(1−SS)}
. | . | . | C
. | Rank . |
---|---|---|---|---|---|---|---|---|---|---|---|

1 | 0.305 | 0.466 | 0.534 | 0.191 | 0.214 | 0.120 | 0.080 | 0.083 | 0.032 | 0.279 | 16 |

2 | 0.407 | 0.359 | 0.641 | 0.255 | 0.198 | 0.160 | 0.074 | 0.054 | 0.057 | 0.514 | 7 |

3 | 0.415 | 0.241 | 0.759 | 0.260 | 0.133 | 0.164 | 0.049 | 0.072 | 0.055 | 0.432 | 11 |

4 | 0.358 | 0.488 | 0.512 | 0.224 | 0.270 | 0.141 | 0.100 | 0.057 | 0.060 | 0.513 | 8 |

5 | 0.498 | 0.386 | 0.614 | 0.312 | 0.213 | 0.196 | 0.079 | 0.035 | 0.092 | 0.728 | 1 |

6 | 0.278 | 0.414 | 0.586 | 0.174 | 0.229 | 0.110 | 0.085 | 0.091 | 0.035 | 0.279 | 16 |

7 | 0.354 | 0.555 | 0.445 | 0.222 | 0.307 | 0.140 | 0.114 | 0.057 | 0.071 | 0.557 | 6 |

8 | 0.322 | 0.456 | 0.544 | 0.202 | 0.252 | 0.127 | 0.093 | 0.072 | 0.048 | 0.397 | 12 |

9 | 0.429 | 0.489 | 0.511 | 0.269 | 0.270 | 0.169 | 0.100 | 0.030 | 0.079 | 0.722 | 2 |

10 | 0.396 | 0.328 | 0.672 | 0.248 | 0.181 | 0.156 | 0.067 | 0.061 | 0.051 | 0.451 | 10 |

11 | 0.434 | 0.408 | 0.592 | 0.272 | 0.225 | 0.171 | 0.084 | 0.039 | 0.071 | 0.644 | 4 |

12 | 0.346 | 0.272 | 0.728 | 0.217 | 0.150 | 0.136 | 0.056 | 0.083 | 0.028 | 0.254 | 18 |

13 | 0.332 | 0.384 | 0.616 | 0.208 | 0.212 | 0.131 | 0.079 | 0.074 | 0.037 | 0.331 | 15 |

14 | 0.314 | 0.467 | 0.533 | 0.197 | 0.258 | 0.124 | 0.096 | 0.075 | 0.049 | 0.394 | 13 |

15 | 0.445 | 0.367 | 0.633 | 0.279 | 0.203 | 0.175 | 0.075 | 0.044 | 0.071 | 0.620 | 5 |

16 | 0.413 | 0.523 | 0.477 | 0.259 | 0.289 | 0.163 | 0.107 | 0.034 | 0.079 | 0.698 | 3 |

17 | 0.36 | 0.438 | 0.562 | 0.226 | 0.242 | 0.142 | 0.090 | 0.059 | 0.052 | 0.468 | 9 |

18 | 0.276 | 0.497 | 0.503 | 0.173 | 0.274 | 0.109 | 0.102 | 0.088 | 0.052 | 0.372 | 14 |

#### Analysis of variance

Relative closeness was used as the response variable. A significance level of 0.05 was selected. The ANOVA was conducted to examine the importance of the six influencing factors on the relative closeness, as shown in Table 11. *R*^{2} (adj) = 87.92% > 80.0% (Table 11) for the influencing factors, indicating that the orthogonal test was statistically significant. Further analysis shows that the differences in the test levels between groups were significant for bend width (*w*), average R-SED height (*h*) and bend centerline radius (*R*) (*P _{w}* < 0.05,

*P*< 0.05 and

_{h}*P*< 0.05). This indicates that these three factors dominated the changing trend of relative closeness. The remaining factors had a test level of

_{R}*P*> 0.05. Thus, these factors showed homogeneity of variance and did not exhibit significant differences. Thus, the remaining factors had insignificant effects on the relative closeness.

Source . | DF . | Adj SS . | Adj MS . | F-value
. | P-value
. | Significance . |
---|---|---|---|---|---|---|

h | 2 | 0.03423 | 0.017113 | 2.24 | 0.012 | * |

s | 2 | 0.01886 | 0.009431 | 1.23 | 0.367 | – |

α | 2 | 0.03362 | 0.016810 | 2.20 | 0.207 | – |

R | 2 | 0.03382 | 0.016911 | 2.21 | 0.015 | * |

w | 2 | 0.23435 | 0.117177 | 15.31 | 0.007 | * |

Q | 2 | 0.01242 | 0.006211 | 0.81 | 0.495 | – |

Error | 5 | 0.03826 | 0.007653 | – | – | – |

Total | 17 | 0.40557 | – | – | – | – |

R^{2} = 0.9057 | R^{2} (adj) = 0.8792 |

Source . | DF . | Adj SS . | Adj MS . | F-value
. | P-value
. | Significance . |
---|---|---|---|---|---|---|

h | 2 | 0.03423 | 0.017113 | 2.24 | 0.012 | * |

s | 2 | 0.01886 | 0.009431 | 1.23 | 0.367 | – |

α | 2 | 0.03362 | 0.016810 | 2.20 | 0.207 | – |

R | 2 | 0.03382 | 0.016911 | 2.21 | 0.015 | * |

w | 2 | 0.23435 | 0.117177 | 15.31 | 0.007 | * |

Q | 2 | 0.01242 | 0.006211 | 0.81 | 0.495 | – |

Error | 5 | 0.03826 | 0.007653 | – | – | – |

Total | 17 | 0.40557 | – | – | – | – |

R^{2} = 0.9057 | R^{2} (adj) = 0.8792 |

**P* < 0.05, and the regression is significant

#### Intuitive analysis

*w*) > average R-SED height (

*h*) > bend centerline radius (

*R*) > R-SED angle (

*α*) > R-SED spacing (

*s*) > discharge flow rate (

*Q*). This shows the degree of influence of each factor on the relative closeness. Thus, for the R-SED design, the effects of the significant factors (bend width (

*w*), average R-SED height (

*h*), bend centerline radius (

*R*)) on the relative closeness should be mainly considered. The values of these significant factors can be selected based on engineering characteristics and the main effect analysis of the relative closeness (Figure 10). The remaining factors (secondary factors) can be selected based on the main effect analysis of the relative closeness (Figure 10) or engineering experience.

Level . | h (cm)
. | s (cm)
. | α (°)
. | R (cm)
. | w (cm)
. | Q (L/s)
. |
---|---|---|---|---|---|---|

1 | 0.4290 | 0.4715 | 0.5410 | 0.4208 | 0.3347 | 0.5125 |

2 | 0.4775 | 0.5242 | 0.4593 | 0.4993 | 0.4943 | 0.4815 |

3 | 0.5357 | 0.4465 | 0.4418 | 0.5220 | 0.6132 | 0.4482 |

Delta | 0.1067 | 0.0777 | 0.0992 | 0.1012 | 0.2785 | 0.0643 |

Rank | 2 | 5 | 4 | 3 | 1 | 6 |

Level . | h (cm)
. | s (cm)
. | α (°)
. | R (cm)
. | w (cm)
. | Q (L/s)
. |
---|---|---|---|---|---|---|

1 | 0.4290 | 0.4715 | 0.5410 | 0.4208 | 0.3347 | 0.5125 |

2 | 0.4775 | 0.5242 | 0.4593 | 0.4993 | 0.4943 | 0.4815 |

3 | 0.5357 | 0.4465 | 0.4418 | 0.5220 | 0.6132 | 0.4482 |

Delta | 0.1067 | 0.0777 | 0.0992 | 0.1012 | 0.2785 | 0.0643 |

Rank | 2 | 5 | 4 | 3 | 1 | 6 |

The main effect plots of relative closeness (*C*) are shown in Figure 10. The regression lines of bend width (*w*), average R-SED height (*h*) and bend centerline radius (*R*) were the steepest, indicating that *C* was the most sensitive to the changes of these three factors. These three factors were positive indicators, i.e., at a larger factor, *C* is larger, and the comprehensive energy dissipation and flow diversion effects of R-SEDs are stronger. *C* was the largest (0.613) at the largest bend width (*w* = 80 cm) while decreasing by 81.90% with *w* decreasing to 50 cm. *C* was the largest (0.537) at the largest average R-SED height (*h* = 1.8 cm) while decreasing by 24.88% with *h* decreasing to 1.2 cm. *C* was the largest (0.524) at the largest bend centerline radius (*R* = 200 cm) while decreasing by 24.76% with *R* decreasing to 140 cm. R-SED angle (*α*) and discharge flow rate (*Q*) were negative indicators, i.e., at a larger factor, *C* is smaller, and the comprehensive energy dissipation and diversion effects of R-SEDs are weaker. *C* reached the maximum (0.542, 0.513) when these two factors were 18° and 20.0 L/s, respectively. R-SED spacing (*s*) was a two-way indicator: a positive indicator within 18–24 cm and a negative indicator within 24–30 cm. At *s* = 24 cm, *C* reached the maximum (0.525). Combined with the findings of Section 3.1.2, the two-way nature of R-SED spacing (*s*) indicates that the value of *s* should not be too large or too small. Excessively large R-SED spacing will reduce the number of R-SEDs, while too small spacing will reduce the contact area of water flow between adjacent R-SEDs. These are not conducive to energy dissipation and flow diversion.

The value of each factor at which the relative closeness reached its maximum was selected to obtain the combination of recommended parameters, i.e., *h* = 1.8 cm, *s* = 24 cm, *α* = 18°, *R* = 200 cm, *w* = 80 cm and *Q* = 20.0L/s.

To verify the accuracy of the multi-criteria evaluation system (i.e., to verify the advantages of the recommended parameter combination) and the necessity of arranging R-SEDs in the bend, numerical simulation was performed in three runs, including (1) Run without R-SEDs (*R* = 200 cm, *w* = 80 cm and *Q* = 20.0 L/s); (2) Run 5 (the optimal run in the 18 orthogonal tests: *h* = 1.5 cm, *s* = 24 cm, *α* = 22°, *R* = 200 cm, *w* = 80 cm and *Q* = 20.0 L/s); (3) the run with the recommended parameter combination (Run OPC) (*h* = 1.8 cm, *s* = 24 cm, *α* = 18°, *R* = 200 cm, *w* = 80 cm and *Q* = 20.0 L/s).

## NUMERICAL SIMULATION OF RECOMMENDED PARAMETER COMBINATIONS

### Water depth variation of the spillway

### Distribution of typical cross-sectional streamlines at the bend

### Vorticity analysis

### Turbulent kinetic energy analysis

*T*is the turbulent kinetic energy (m

^{2}/s

^{2});

*u’*,

*v’*and

*w’*are the pulsatile flow velocity in the longitudinal, transverse and vertical directions (m/s), respectively.

*Z,*and the maximum value was concentrated at the bend bottom. This is because turbulent shear stress at the solid-liquid interface was larger, resulting in larger turbulent energy. Thus, swirling eddies on different scales can be easily generated. The upward movement and mixing of the vortexes were accompanied by energy transfer and dissipation. Thus, this resulted in smaller turbulent energy in the upper flow (>0.5

*Z*). In Run without R-SEDs, the distribution of turbulent kinetic energy in each cross-section of the bend was similar. The turbulent energy reached the maximum at the inlet of the bend (0.183 m

^{2}/s

^{2}) and then gradually decreased. After the arrangement of R-SEDs at the bend bottom (Run 5 and Run OPC), the contact area between the flows was increased by the discharge flow hitting the R-SEDs. This enhanced the overall turbulence of the flow, and the turbulent kinetic energy below 0.5

*Z*increased. Most turbulent kinetic energy and the maximum value of each cross-section below 0.5

*Z*in Run OPC were larger than in Run 5. The maximum value in Run OPC (0.334 m

^{2}/s

^{2}) (Figure 14(h)) increased by 82.51 and 32.02% compared to those in Run without R-SEDs (0.183 m

^{2}/s

^{2}) and Run 5 (0.253 m

^{2}/s

^{2}), respectively. In Figure 14(a)–14(d), the turbulent kinetic energy curves of Run 5 and Run OPC showed some overlaps. In Figure 14(f)–14(i), the differences between the turbulent kinetic energy of these two runs below 0.5

*Z*became significant. This can be explained according to the findings in Section 4.2, i.e., when the water reached the middle cross-section of the bend (#g, see Figure 14(e)), the R-SEDs in Run OPC allowed the circulation to more fully developed. Under the combined effects of centrifugal force, water pressure and R-SED energy dissipation and flow diversion, the shear stress in the side wall of both spillway banks increased in Run OPC. The mixing of upper and lower flows was enhanced, resulting in a higher turbulent diffusion rate and, thus, a higher turbulence energy growth rate.

### Comparison of the results of energy dissipation rate and water surface superelevation coefficient

The above numerical results show that R-SEDs' arrangement in the spillway bend can facilitate good energy dissipation and flow diversion. All the hydraulic indexes in Run OPC were superior to those in Run 5. These verified the reliability of the established multi-criteria evaluation system (i.e., the advantages of the recommended parameter combination) and the necessity of arranging R-SEDs in the bend.

## CONCLUSIONS

This study focused on the R-SEDs arranged at the bend bottom of the curved spillway. A multi-criteria evaluation system was established for the comprehensive energy dissipation and flow diversion effects of R-SEDs. Based on hydrodynamic theory, orthogonal tests and numerical simulation, the six factors (average R-SED height *h*, R-SED spacing *s,* R-SED angle *α*, bend width *w*, bend centerline radius *R* and discharge flow rate *Q*) affecting the energy dissipation and diversion effects of the R-SEDs were analyzed to obtain an optimal parameter combination. The main conclusions are drawn as follows:

- (i)
Bend width (

*w*) and bend centerline radius (*R*) are the significant factors affecting the energy dissipation effect of R-SEDs. These two factors are positively correlated with the energy dissipation rate. The factors are ranked in descending order in terms of their effects on R-SEDs' energy dissipation performance: bend width (*w*) > bend centerline radius (*R*) > R-SED angle (*α*) > R-SED spacing (*s*) > discharge flow rate (*Q*) > average R-SED height (*h*). - (ii)
Average R-SED height (

*h*), R-SED spacing (*s*) and bend centerline radius (*R*) are the significant factors affecting the flow diversion effect of R-SEDs. Average R-SED height (*h*) is negatively correlated with the water surface superelevation coefficient, while R-SED spacing (*s*) and bend centerline radius (*R*) are positively correlated. The factors are ranked in descending order in terms of their effects on R-SEDs' flow diversion performance: average R-SED height (*h*) > bend centerline radius (*R*) > R-SED spacing (*s*) > R-SED angle (*α*) > bend width (*w*) > discharge flow rate (*Q*). - (iii)
Bend width (

*w*), average R-SED height (*h*) and bend centerline radius (*R*) are the significant factors influencing the comprehensive energy dissipation and flow diversion effects. These three factors are positively correlated with relative closeness. The factors are ranked in descending order in terms of their effects on R-SEDs' comprehensive performance: bend width (*w*) > average R-SED height (*h*) > bend centerline radius (*R*) > R-SED angle (*α*) > R-SED spacing (*s*) > discharge flow rate (*Q*). - (iv)
R-SEDs are simple in shape and convenient in construction. They can effectively improve the various undesirable hydraulic phenomena in the bend flow. The influence weights of the energy dissipation rate (0.629) and the water surface superelevation coefficient (0.371) indicate that the energy dissipation effect of the R-SEDs on the bend flow is more significant than the flow diversion effect.

- (v)
Among the 18 groups of orthogonal tests, Run 5 shows the highest overall rating. The run with the recommended parameter combination (

*h*= 1.8 cm,*s*= 24 cm,*α*= 18°,*R*= 200 cm,*w*= 80 cm and*Q*= 20.0 L/s) from the multi-criteria evaluation system exhibits better hydraulic evaluation indexes than Run 5. For example, the energy dissipation rate (0.591) is 18.67% higher than that in Run 5. The water surface superelevation coefficient (0.306) is 26.14% lower than that in Run 5. The recommended parameter combination can provide a reference for the R-SED design of similar curved spillways based on actual project topography and geological conditions. - (vi)
This study mainly considers the effects of the six key factors (average R-SED height

*h*, R-SED angle*α*, R-SED spacing*s*, bend width*w*, bend centerline radius*R*and discharge flow rate*Q*) on the energy dissipation and flow diversion effects of R-SEDs. Further studies on other factors (such as bend angle and bottom slope) will be conducted subsequently. Regarding the R-SED design of the same type of curved spillways, the effects of relevant parameters on R-SEDs' energy dissipation and flow diversion effects in this study can provide a reference for selecting the parameter values. For example, based on the recommended parameter values, the main effect analysis results and engineering characteristics, relevant simulation tests can be performed to select the values of significant factors. For the secondary factors, combined with the recommended parameter values and the main effect analysis results, their values can be directly taken based on engineering experience or economic practicality. The relevant parameters used in this study were the dimensions of the physical laboratory model. Thus, these parameters should be scaled up according to the geometric scale (1:50) for actual engineering design.

## ACKNOWLEDGEMENTS

The authors gratefully acknowledge the financial support from the National Natural Science Foundation of China (Grant numbers 52269019 and 51769037) and the University Research Program Innovation Team Project of Xinjiang Uygur Autonomous Region (Grant number XJEDU2017T004). We would also like to thank XSG Editing for their professional editing service.

## FUNDING

This work was supported by the National Natural Science Foundation of China (Grant numbers 52269019 and 51769037) and University Research Program Innovation Team Project of Xinjiang Uygur Autonomous Region (Grant number XJEDU2017T004). The first author has received research support from Xinjiang Agricultural University.

## DATA AVAILABILITY STATEMENT

All relevant data are included in the paper or its Supplementary Information.

## CONFLICT OF INTEREST

The authors declare there is no conflict.